Past exam of the advanced quantum physics course of the University of Cambridge / 2021 / Section B / 5 / a / Solution

The components of the momentum operator do not generally commute with a position-dependent magnetic vector potential. Expanding minimal coupling with the ordering intact givesActing on a wavefunction shows that $\mathbf{p}\cdot \mathbf{A}-\mathbf{A}\cdot \mathbf{p}=-i\hslash \nabla \cdot \mathbf{A}$. In the Coulomb gauge the divergence vanishes, so the two cross terms agree. Discarding the quadratic term in the weak radiation amplitude givesThe equality of the two operator orderings follows from the gauge condition, not from treating $\mathbf{p}$ and $\mathbf{A}$ as commuting numbers component by component.